If \(\lim\limits_{n \to \infty } {\large\frac{{{a_{n + 1}}}}{{{a_n}}}\normalsize} = 1,\) then the series \(\sum\limits_{n = 1}^\infty {{a_n}}\) may converge or diverge and the ratio test is inconclusive; some other tests must be used.

The Root Test

Consider again the series \(\sum\limits_{n = 1}^\infty {{a_n}}\) with positive terms. According to the Root Test:

If \(\lim\limits_{n \to \infty } \sqrt[\large n\normalsize]{{{a_n}}} = 1,\) then the series \(\sum\limits_{n = 1}^\infty {{a_n}},\) may converge or diverge, but no conclusion can be drawn from this test.